\[ \int (2+e x)^{3/2} \sqrt {12-3 e^2 x^2} \, dx \]
Optimal antiderivative \[ -\frac {32 \left (-e x +2\right )^{\frac {3}{2}} \sqrt {3}}{3 e}+\frac {16 \left (-e x +2\right )^{\frac {5}{2}} \sqrt {3}}{5 e}-\frac {2 \left (-e x +2\right )^{\frac {7}{2}} \sqrt {3}}{7 e} \]
command
integrate((e*x+2)^(3/2)*(-3*e^2*x^2+12)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2}{105} \, \sqrt {3} {\left (84 \, {\left (x e - 2\right )}^{2} \sqrt {-x e + 2} + {\left ({\left (15 \, {\left (x e - 2\right )}^{3} \sqrt {-x e + 2} + 84 \, {\left (x e - 2\right )}^{2} \sqrt {-x e + 2} - 140 \, {\left (-x e + 2\right )}^{\frac {3}{2}}\right )} e^{\left (-2\right )} + 352 \, e^{\left (-2\right )}\right )} e^{2} - 420 \, {\left (-x e + 2\right )}^{\frac {3}{2}} + 672\right )} e^{\left (-1\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \sqrt {-3 \, e^{2} x^{2} + 12} {\left (e x + 2\right )}^{\frac {3}{2}}\,{d x} \]________________________________________________________________________________________